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This study comprehensively explores the (3 + 1) -dimensional Mikhailov–Novikov–Wang (ℕ) integrable equation, with the primary objective of elucidating its physical manifestations and establishing connections with analogous nonlinear evolution equations. The investigated model holds significant physical meaning across various disciplines within mathematical physics. Primarily, it serves as a fundamental model for understanding nonlinear wave propagation phenomena, offering insights into wave behaviors in complex media. Moreover, its relevance extends to nonlinear optics, where it governs the dynamics of optical pulses and solitons crucial for optical communication and signal processing technologies. Employing analytical methodologies, namely the unified () , Khater II ( hat.II) method, and He's variational iteration (ℍ ) method, both numerical and analytical solutions are meticulously examined. Through this investigation, the intricate behaviors of the equation are systematically unveiled, shedding illuminating insights on various physical phenomena, notably including wave propagation in complex media and nonlinear optics. The outcomes not only underscore the efficacy of the analytical techniques deployed but also accentuate the equation's pivotal role in modeling a broad spectrum of nonlinear wave dynamics. Consequently, this research significantly advances our comprehension of complex physical systems governed by nonlinear dynamics, thereby contributing notably to interdisciplinary pursuits in mathematical physics. [ABSTRACT FROM AUTHOR]

The nonlinear Semilinear Dispersive-Fisher (SDF) model is an important equation describing wave dynamics in dispersive media influenced by nonlinear and diffusive effects. This study enhances both analytical and numerical methods to solve the SDF model and validate the solutions. We utilize the Modified Khater (MKhat) and Unified (UF) methods for analytical solutions, and He's Variational Iteration (HVI) scheme for numerical approximations. Our investigation connects the SDF model to other nonlinear equations like the Korteweg–de Vries (KdV) and Fisher–Kolmogorov (FK) equations, demonstrating the consistency between analytical and numerical solutions. This research contributes to the understanding and modeling of wave phenomena in dispersive media with nonlinear and diffusive dynamics, offering refined analytical and numerical techniques specific to the SDF model. Key contributions include introducing the MKhat and UF methods for analytical solutions and employing the HVI scheme for numerical approximations, which are less represented in current literature. This work is positioned in mathematical physics, focusing on nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This study aims to solve the nonlinear Pochhammer–Chree (ℕℙℂ) equation to understand its physical implications and establish connections with other nonlinear evolution equations, particularly in plasma dynamics. By using the Khater II (핂hat. II) method for analytical solutions and validating these solutions numerically with the variational iteration method, this study offers a detailed understanding of the equation’s behavior. The results demonstrate the effectiveness of these methods in accurately modeling the system, highlighting the importance of combining analytical and numerical approaches for reliable solutions. This research significantly advances the field of nonlinear dynamics, especially in plasma physics, by employing multidisciplinary methods to tackle complex physical processes. Moreover, the ℕℙℂ equation is relevant in various physical contexts beyond plasma dynamics such as optical and quantum fields. In optics, it models the propagation of nonlinear waves in fiber optics, where similar nonlinear evolution equations describe wave interactions. In quantum field theory, the ℕℙℂ equation helps in understanding the behavior of quantum particles and fields under nonlinear effects, making it a versatile tool for studying complex phenomena across different domains of physics. [ABSTRACT FROM AUTHOR]

This study aims to investigate the modified nonlinear time fractional Harry Dym equation using analytical and numerical techniques. The modified nonlinear time fractional Harry Dym equation is a generalization of the classical Harry Dym equation, which describes the propagation of nonlinear waves in a variety of physical systems. The conformable fractional derivative is used to define the time fractional derivative in the equation, which provides a natural and straightforward approach. The Khater II method, a powerful analytical technique, is employed to obtain approximate solutions for the equation. Additionally, three numerical schemes, namely, Cubic-B-spline, Quantic-B-spline and Septic-B-spline schemes, are developed and implemented to solve the equation numerically. The numerical results are compared with other numerical solutions to assess the accuracy and efficiency of the proposed schemes. The physical meaning of the modified nonlinear time fractional Harry Dym equation is discussed in detail, and its relation to other nonlinear evolution equations is highlighted. The results of this study provide new insights into the behavior of nonlinear waves in physical systems and contribute to a better understanding of the physical characterizations of the modified nonlinear time fractional Harry Dym equation. [ABSTRACT FROM AUTHOR]

This study explores the nonlinear Peyrard-Bishop DNA dynamic model, a nonlinear evolution equation that describes the behavior of DNA molecules by considering hydrogen bonds between base pairs and stacking interactions between adjacent base pairs. The primary objective is to derive analytical solutions to this model using the Khater III and improved Kudryashov methods. Subsequently, the stability of these solutions is analyzed through Hamiltonian system characterization. The Peyrard-Bishop model is pivotal in biophysics, offering insights into the dynamics of DNA molecules and their responses to external forces. By employing these analytical techniques and stability analysis, this research aims to enhance the understanding of DNA dynamics and its implications in fields such as drug design, gene therapy, and molecular biology. The novelty of this work lies in the application of the Khater III and an enhanced Kudryashov methods to the Peyrard-Bishop model, along with a comprehensive stability investigation using Hamiltonian system characterization, providing new perspectives on DNA molecule dynamics within the scope of nonlinear dynamics and biophysics [ABSTRACT FROM AUTHOR]

This study investigates the nonlinear time-fractional Harry Dym ( ℍ ) equation, a model with significant applications in soliton theory and connections to various other nonlinear evolution equations. The Harry Dym (ℍ ) equation describes the propagation of nonlinear waves in various physical contexts, including shallow water waves, nonlinear optics, and plasma physics. The fractional-order derivative introduces a memory effect, allowing the model to capture nonlocal interactions and long-range dependencies in the wave dynamics. The primary objective of this research is to obtain accurate analytical solutions to the ℍ equation and explore its physical characteristics. We employ the Khater III method as the primary analytical technique and utilize the He's variational iteration (ℍ ) method as a numerical scheme to validate the obtained solutions. The close agreement between analytical and numerical results enhances the applicability of the solutions in practical applications of the model. This research contributes to a deeper understanding of the ℍ equation's behavior, particularly in the presence of fractional-order dynamics. The obtained solutions provide valuable insights into the complex interplay between nonlinearity and memory effects in the wave propagation phenomena described by the model. By shedding light on the physical characteristics of the ℍ equation, this study paves the way for further investigations into its potential applications in diverse physical settings. [ABSTRACT FROM AUTHOR]

The objective of this paper is to examine the analytical properties of the nonlinear (1+1)-dimensional Camassa–Holm equation (ℂℍ), a fundamental model within the domain of nonlinear evolution equations. The aforementioned equation serves as a valuable tool in elucidating the unidirectional propagation of shallow water waves over a level terrain, as well as in representing certain nonlinear wave phenomena seen in cylindrical hyperelastic rods. We use the Khater III (핂hat.III) and improved Kudryashov (핀핂ud) technique to provide accurate solutions, drawing inspiration from the intricate mathematical framework of the ℂℍ issue. He’s variational iteration (ℍ핍핀) technique is used as a numerical methodology to assess the correctness of the generated answers. This strategy reveals a notable concurrence between the analytical and numerical outcomes. This alignment guarantees the suitability of the acquired solutions within the framework of the studied model.The importance of this investigation lies in its ability to improve our comprehension of the intricate dynamics regulated by the ℂℍ equation and its connections with other nonlinear evolution equations that describe shallow water wave behaviors and nonlinear wave propagation in cylindrical hyperelastic rods. The results of the study demonstrate novel analytical approaches, expanding the range of potential solutions and offering valuable insights into the physical characteristics of the interconnected wave phenomena. This research offers valuable insights and methodologies for addressing intricate mathematical models in shallow water wave theory and studying nonlinear waves in hyperelastic materials, therefore making substantial advances to the subject of nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

This study delves into the perturbed Radhakrishnan–Kundu–Lakshmanan ( (p RKL) ) model, a pivotal component within nonlinear optics and communication engineering. In addressing this challenge, we employed the Bernoulli sub-equation function method and the Khater II method as analytical approaches, complemented by numerical solutions authenticated through the exponential cubic B–spline (ECBS) method. Our investigation yielded innovative solitary wave solutions, depicted elegantly through three-dimensional, two-dimensional, and contour plots. To ascertain the precision and dependability of these outcomes, we conducted a comparative analysis between analytical and numerical findings, visually presenting them via two-dimensional graphs. The implications of our discoveries are significant, offering insights into perturbations of optical solitons within nonlinear optical fibers. Our research effectively concludes that the proposed methodologies successfully solve the p RKL model, yielding highly accurate solutions. Notably, our study introduces novelty to nonlinear optics by applying the Bernoulli sub-equation function (BSE) method, the Khater II (Kh II) method, and the exponential cubic B-spline (ECBS) method to the p RKL model, with a specific focus on solitary wave solutions. [ABSTRACT FROM AUTHOR]

In this work, three models of the nonlinear Schrödinger equation are looked at to see if traveling wave solutions have unique structures. "The generalized Korteweg–de Vries, (2+1)-dimensional Ablowitz–Kaup–Newell–Segur, and the Maccari models" are among the examined systems. The kinetic energy operator is affected by where the mass is, how it changes over time, and how the kinetic energy operator is shown. Using the generalized Riccati expansion method, new soliton wave solutions are constructed for the models that have been looked at. Several different graphics depict the numerical simulations of the deduced answers. Verifying and improving these methods (Mathematica 13.1) means checking to see if they are right and re-entering the results into the models. [ABSTRACT FROM AUTHOR]

This study investigates novel solitary wave solutions of the Gilson–Pickering ( ℙ) equation, which is a model that describes the motion of a fluid in a porous medium. An analytical scheme is applied to construct these solutions, utilizing the extended Khater method in conjunction with the homogenous balance technique. The derived expressions for the solitary wave solutions are exact and are presented in terms of hyperbolic functions. The ℙ equation is valuable for a wide range of applications, including oil and gas reservoir engineering, groundwater flow, and flow in biological tissues. Additionally, this model is employed to describe the behavior of waves in various physical systems such as fluids and plasmas. Specifically, it models the propagation of dispersive waves in a media that exhibits both dispersion and dissipation. To ensure the accuracy of the constructed solutions, a numerical scheme is employed. The properties of the solitary wave solutions are analyzed, and their physical implications are explored. The results of this investigation reveal a rich variety of solitary wave solutions that exhibit interesting behaviors, including oscillatory and non-oscillatory behavior, which are elucidated through various types of distinct graphs. Consequently, this study provides significant insights into the behavior of fluid flow in porous media and its applications in various fields, including oil and gas reservoir engineering and groundwater flow modeling. The analytical and numerical methods employed in this investigation demonstrate their potential for studying nonlinear evolution equations and their applications in the physical sciences. [ABSTRACT FROM AUTHOR]

This study uses crystal lattice theory and physicochemical characterization to show a number of correct wave solutions that are like the way plasma waves move. The nonlinear time–fractional Gilson–Pickering (G P) model has been addressed using two distinct analytical and numerical techniques. This model is used in crystal lattice theory and plasma physics to show how waves move, so it is a basic model for how waves move in one direction. Utilizing the modified rational and He's variational iteration approximations in conjunction with the β–fractional derivative principles, we give the handled model fresh and precise solitary wave solutions. Various contour, three-dimensional, and two-dimensional graphs depict the determined outcomes. Compared to other recent studies, ours indicates the importance of this research area. The presented methods show how simple, direct, and effective they are and how they can be used with a wide range of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This paper investigates the soliton wave solution of the nonlinear biological population (NBP) model by employing a novel computational scheme. The selected model for this study describes the logistics of the population because of births and deaths. Some novel structures of the NBP model's solutions, are obtained such as exponential, trigonometric, and hyperbolic. These solutions are clarified through some distinct graphs in contour three plot, three-dimensional, and two-dimensional plots. The Hamiltonian system's characterizations are used to check the obtained solutions' stability. The solutions' accuracy is checked by handling the NBP model through the variational iteration (VI) method. The matching between analytical and semi-analytical solutions shows the accuracy of the obtained solutions. The method's performance shows its effectiveness, power, and ability to apply to many nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This paper explores accurate, stable, and novel soliton wave solutions of the nonlinear Qiao model. This model, which was derived in 2007, possesses a Lax representation and bi-Hamiltonian structure, where it is a second positive member in the utterly integrable hierarchy. The well-known generalized extended tanh-function method is employed to construct novel soliton wave solutions. The stable property of the obtained solutions is examined along with the Hamiltonian system's characterizations. Furthermore, the accuracy of the obtained solutions is checked by comparing it with the model's semi-analytical solutions that have been obtained by employing the variational iteration (VI) method. The obtained analytical and semi-analytical solutions are demonstrated through some distinct graphs to show more physical and dynamical behavior of the investigated model. The used analytical and semi-analytical schemes' performance is checked to show if it is effective and powerful. [ABSTRACT FROM AUTHOR]

In this research paper, the modified Khater method, the Adomian decomposition method, and B-spline techniques (cubic, quintic, and septic) are applied to the deoxyribonucleic acid (DNA) model to get the analytical, semi-analytical, and numerical solutions. These solutions comprise much information about the dynamical behavior of the homogenous long elastic rods with a circular section. These rods constitute a pair of the polynucleotide rods of the DNA molecule which are plugged by an elastic diaphragm that demonstrates the hydrogen bond's role in this communication. The stability property is checked for some solutions to show more effective and powerful of obtained solutions. Based on the role of analytical and semi-analytical techniques in the motivation of the numerical techniques to be more accurate, the B-spline numerical techniques are applied by using the obtained exact solutions on the DNA model to show which one of them is more accurate than other, to explain more of the dynamic behavior of the homogenous long elastic rods, and to show the coincidence between the different types of obtained solutions. The obtained solutions verified with Maple 16 & Mathematica 12 by placing them back into the original equations. The performance of these methods shows the power and effectiveness of them for applying to many different forms of the nonlinear evolution equations with an integer and fractional order. [ABSTRACT FROM AUTHOR]

In this paper, the solitary wave solutions of the complex perturbed Gerdjikov–Ivanov (CPGI) equation are investigated by employing the extended simplest equation (ESE) and modified Khater (MKhat) methods. The studied equation describes the physical characterization of the optical soliton waves to mitigate internet bottlenecks with many different applications in the telecommunication industry. The evaluated solutions are explained through various sketches in two-, three-dimensional, and contour plots of their real, imaginary, and absolute values. The novelty of the obtained results is demonstrated by comparing them with the previously obtained solutions. The computational applied schemes' performance is tested to illustrate their powerful and effective handling of many nonlinear evolutions equations. [ABSTRACT FROM AUTHOR]

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified. [ABSTRACT FROM AUTHOR]

This paper investigates the high-frequency waves' dynamical behavior in the relaxation medium through two recent analytical schemes. This study depends on the Vakhnenko–Parkes (VP) equation that has been reduced from the well-known Ostrovsky equation. The modified Khater (MKhat) and the extended simplest equation (ESE) methods are used to handle the considered model. As a result, many novel solitary wave solutions have been obtained to construct the initial and boundary conditions. These conditions allow employing the variational iteration (VI) method to study the semi-analytical solutions of the considered model. The accuracy of solutions is explained along with showing the matching between analytical and semi-analytical solutions and comparing our obtained solutions with the previous results that have been obtained in published research papers. Moreover, the high-frequency waves' behavior relaxation medium is illustrated through some distinct sketches. The methods' performance shows their effectiveness, direct, easy, and consequential for studying many nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical solutions of the perturbed nonlinear Schrödinger equation through the modified Khater method. This method is considered one of the most recent accurate analytical schemes in nonlinear evolution equations where it obtained many distinct forms of solutions of the considered model. The investigated model in this paper is an icon in quantum fields where it describes the wave function or state function of a quantum-mechanical system. The physical characterization of some obtained solutions in our study is explained through sketching them in two- and three-dimensional contour plots. The novelty of our study is clear by showing the matching between our solutions and those that have been constructed in previously published papers. [ABSTRACT FROM AUTHOR]

This article studies the generalized (2 + 1)-dimensional shallow water equation by applying two recent analytical schemes (the extended simplest equation method and the modified Kudryashov method) for constructing abundant novel solitary wave solutions. These solutions describe the bidirectional propagating water wave surface. Some obtained solutions are sketched in two- and three-dimensional and contour plots for demonstrating the dynamical behavior of these waves along shallow water. The accuracy of the obtained solutions and employed analytical schemes is investigated using the evaluated solutions to calculate the initial condition, and then the well-known variational iterational (VI) method is applied. The VI method is one of the most accurate semi-analytical solutions, and it can be applied for high derivative order. The used schemes' performance shows their effectiveness and power and their ability to handle many nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this article, two recent computational schemes [the modified Khater method and the generalized exp − φ (I) –expansion method] are applied to the nonlinear predator–prey system for constructing novel explicit solutions that describe a prototype of an excitable system. Many distinct types of solutions are obtained such as hyperbolic, parabolic, and rational. Moreover, the Hamiltonian system's characteristics are employed to check the stability of the obtained solutions to show their ability to be applied in various applications. 2D, 3D, and contour plots are sketched to illustrate more physical and dynamical properties of the obtained solutions. Comparing our obtained solutions and that obtained in previous published research papers shows the novelty of our paper. The performance of the two used analytical schemes explains their effectiveness, powerfulness, practicality, and usefulness. In addition, their ability in employing various forms of nonlinear evolution equations is also shown. [ABSTRACT FROM AUTHOR]

This research paper employs two different computational schemes to the couple Boiti–Leon–Pempinelli system and the (3+1)-dimensional Kadomtsev–Petviashvili equation to find novel explicit wave solutions for these models. Both models depict a generalized form of the dispersive long wave equation. The complex, exponential, hyperbolic, and trigonometric function solutions are some of the obtained solutions by using the modified Khater method and the Jacobi elliptical function method. Moreover, their stability properties are also analyzed, and for more interpretation of the physical features of the obtained solutions, some sketches are plotted. Additionally, the novelty of our paper is explained by displaying the similarity and difference between the obtained solutions and those obtained in a different research paper. The performance of both methods is tested to show their ability to be applied to several nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This research paper applies the modified Khater method and the generalized Kudryashov method to the general Degasperis–Procesi (DP) equation, which is used to describe the dynamical behavior of the shallow water outflows. Some shock peakon wave solutions are obtained by using these methods. Moreover, some figures are sketched for these solutions to explain more physical properties of the general DP equation and to figure out the coincidence between different types of obtained solutions. The stability property by using the features of the Hamiltonian system is tested to some obtained solutions to show their ability for applying in the model's applications. The obtained solutions were verified with Maple 16 & Mathematica 12 by placing them back into the original equations. The performance of these methods shows their power and effectiveness for applying to many different forms of the nonlinear evolution equations with an integer and fractional order. [ABSTRACT FROM AUTHOR]

The accuracy of novel lump solutions of the potential form of the three–dimensional potential Yu–Toda–Sasa–Fukuyama (3-Dp-YTSF) equation is investigated. These solutions are obtained by employing the extended simplest equation (ESE) and modified Kudryashov (MKud) schemes to explore its lump and breather wave solutions that characterizes the dynamics of solitons and nonlinear waves in weakly dispersive media, plasma physics, and fluid dynamics. The accuracy of the obtained analytical solutions is investigated through the perspective of numerical and semi-analytical strategies (septic B-spline (SBS) and variational iteration (VI) techniques). Additionally, matching the analytical and numerical solutions is represented along with some distinct types of sketches. The superiority of the MKud is showed as the fourth research paper in our series that has been beginning by Mostafa M. A. Khater and Carlo Cattani with the title "Accuracy of computational schemes". The functioning of employed schemes appears their effectual and ability to apply to different nonlinear evolution equations. [ABSTRACT FROM AUTHOR]